ZDM 2005 Vol. 37 (5)
Balancing Mathematics Education
Research and the NCTM Standards
Bharath Sriraman (USA)
Michelle Pizzulli (USA)
Abstract: The release of the Principles and Standards for
School Mathematics in the United States by the National
Council of Teachers of Mathematics (NCTM) brought to the
forefront the debate of whether research should determine the
validity of the espoused Standards? Or conversely whether the
Standards should influence the research agenda of the
mathematics education community? How should university
teacher educators address this issue? Should pre-service and
practicing teachers blindly accept the Standards as well as the
research, or do we cultivate the critical thinking skills that will
allow preparing teachers to resolve this dilemma? In this article
a university mathematics educator and an idealistic pre-service
elementary teacher try to resolve the dilemma of balancing the
Standards with research and personal beliefs about the teaching
and learning of mathematics.
Kurzreferat: Die Veröffentlichung der Prinzipien und
Standards für Schulmathematik in den Vereinigten Staaten
durch den National Council of Teachers of Mathematics
(NCTM) brachte im Vorfeld die Debatte ins Rollen, ob die
Forschung die Gültigkeit der befürworteten Standards
determinieren sollte. Oder im Gegenteil, ob die Standards die
Forschungsagenda der Mathematikdidaktiker beeinflussen
sollte. Wie würden die universitären Ausbilder von
Mathematiklehrern mit dieser Fragestellung umgehen? Sollten
zukünftige und bereits praktizierende Lehrer diese Standards
ebenso blind akzeptieren wie die Forschung, oder kultivieren
wir kritisches Denken, welches die Lehrer darauf vorbereitet,
dieses Dilemma zu lösen? In diesem Artikel versuchen ein
universitärer Mathematiklehrerausbilder und ein idealistischer
Lehrer in der Ausbildung dieses Dilemma zu lösen, indem sie
die Standards mit der Forschung und persönlichen Beliefs über
das Lehren und Lernen von Mathematik ausbalancieren.
ZDM-Classification: C60
Introduction
The Principles and Standards for School Mathematics 1,
a publication of the National Council of Teachers of
Mathematics (NCTM, 2000), provides an ideal vision of
the mathematical content and processes that are relevant
in the 21st century. The strand for elementary students
revolves around five overarching basic goals, which are
to help students cultivate number sense, methods of
estimation and measurement, pattern recognition,
geometric concepts, and statistical methods. It is laudable
that these overall goals remain the same as a student
progresses from the early elementary level to the late high
school level while the depth of investigation within a
particular content domain increases. It makes perfect
sense to introduce concepts early in an intuitive manner,
and to build on these as students’ reasoning abilities
1
Referred to as the Standards in the remainder of the
paper.
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Analyses
mature. The NCTM sets high standards for students at all
levels to measure up to. The agenda set for the
elementary level (Kindergarten – Grade 5) is challenging
especially in the realm of geometry where students are
expected to construct convincing arguments and proofs to
solve problems and/or draw conclusions about geometric
figures and patterns. In essence by placing primary
responsibility for establishing the validity of
mathematical ideas with the students, the Standards are
viewing each student as a mathematician. Having
students justify the validity of their ideas promotes the
importance of being able to communicate with others by
organizing arguments in a coherent way. How are
teachers, especially those that are new to the profession
supposed to achieve the high expectations on learning
outcomes set by the Standards? Shouldn’t classroom
pedagogy also take into account the current research on
teaching and learning? Are the Standards and Research
mutually exclusive? Or can they be harmoniously
balanced? These are the topics of exploration in this
paper.
A Pragmatic View of Standards and Research
Standards are statements about what is most valued, so
they need not be justified by research although some of
the statements are backed by research (Hiebert, 1999).
Most mathematics educators’ position is that research can
inform this debate but is not a necessary condition.
Research cannot prove what is “best”. Human judgement
has to decide what is “best”. The best example is the use
of calculators in the classroom. Some of the most
brilliant ideas in mathematics and the sciences have come
out of intuition. Hiebert (1999) says that the research
process can help one see things differently and imagine
new possibilities. Research is helpful in documenting the
effectiveness or the ineffectiveness of new ideas and can
suggest explanations for successes and failures. Research
can help document the current state of teaching in the
classroom, the curricular materials being used, and how
students are learning.
One of the jokes among mathematics educators in the US
is: if Benjamin Franklin were to miraculously appear in
the world today, he would be unable to cope with all the
changes that have occurred. However he would surely
recognize a classroom because of the consistent way in
which school subjects are taught! Research in
mathematics education has shown that the mode of
instruction has been consistent even in the face of reform.
The routine followed in most traditional mathematics
classrooms are: solutions to the previous days homework,
short presentation of the next topic, followed by seat
work, during which the teacher moves about the
classroom answering questions (Fey, 1979; Hiebert,1999
; Stigler & Hiebert, 1997). Emphasis on procedures and
the classroom routine just mentioned are the defining
characteristics of most mathematics classrooms. The
history of reform in the United States has shown that the
main reasons why promising alternative programs fail
are: (1) They aren’t implemented effectively when
adopted by schools and districts; (2) They are not given
enough time to truly assess whether they are successful or
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not; (3) Lack of teacher preparation., and (4) Lack of
documentation of how alternative programs are being
implemented in the classroom.
The Hidden Role of Beliefs
New programs expect teachers to make over night
changes in their mode of classroom instruction. Reform
does not happen readily, since teachers look through old
lenses, and if teachers’ beliefs don’t change, neither will
their methods. In subscribing to a constructivist view of
mathematics teaching and learning the NCTM Standards
envisions classrooms in which students learn
mathematics as they construct ways of dealing with
problematic situations. They do this by reflecting on their
interactions with the world and their peers, and their
attempts to make sense of those interactions. This vision
of the Standards is unlikely to become reality if teachers
still view themselves in the traditional role of transmitter
(and authorities) of knowledge.
Ernest (1989) categorized three philosophies of
mathematics, namely the instrumentalist view, the
Platonist view, and the problem solving view. The
instrumentalist sees mathematics as a collection of facts
and procedures, which have utility. The Platonist sees
mathematics as a static but unified body of knowledge.
Mathematics is discovered, not created. The problem
solving view looks on mathematics as continually
expanding and yet lacking ontological certainty. The
problem solving view sees mathematics as a cultural
artifact. This implies that what is thought as true today,
may not be seen as true tomorrow (pp. 99-199). Ernest
also describes absolutist and fallibilist views of
mathematical certainty. The absolutist sees mathematics
as completely certain and the fallibilist recognizes that
mathematical truth may be challenged and revised
(Ernest, 1991, p.3). Lerman (1990) recognizes that ones
philosophy is related to ones preferred teaching style. The
absolutist teacher will prefer a direct teaching style
whereas a fallibilist is much more likely to engage in
exploratory activities and open-ended problems.
Thompson (1984) studied three middle school teachers,
all of whom had different beliefs about the nature of
mathematics. The first teacher viewed mathematics as a
coherent collection of interrelated concepts and
procedures. She regarded mathematics as a subject free of
ambiguity and emphasized conceptual development in the
students. She would fit Ernest’s model as a Platonist.
The second teacher had a very different perspective of
mathematics and her teaching reflected more of a
process-oriented approach than a content oriented
approach. A view of mathematics as a subject that allows
for the discovery of properties and relationships through
personal inquiry seemed to underlie her instructional
approach. This teacher viewed problem solving as the
fundamental goal of learning and regarded mathematical
notation as arbitrary. She would fit Ernest’s model as
person with the problem solving view. The third teacher
in Thompson’s study saw mathematics as a collection of
facts and procedures, which help students’ find the
answer. She saw no ambiguity in mathematics. She
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ZDM 2005 Vol. 37 (5)
would fit Ernest’s model as a person with an
instrumentalist view. Thompson sees at least three
distinct ways of viewing mathematics, all of which
greatly influence the choice of curriculum and its
delivery. Thompson (1992) says that research on teachers
cognitions, studies of teachers’ conceptions have
contributed to a conceptual shift in the field of research
on teaching, moving away from a behavioral conception
of teaching towards “a conception that takes account of
teachers as rational beings” (p.142). Our understanding of
teaching from teachers’ perspectives complements our
growth of understanding of learning from learners’
perspectives, which in turn, enriches the idea of schooling
as the negotiation of norms, practices and meanings
(Cobb, 1988).
Implications for Elementary School Mathematics
Educators
In the U.S, many pre-service elementary teachers take a
two-semester sequence of mathematics content courses at
the university level. This is followed by generic methods
courses. As a mathematics educator at the university, one
has the heavy responsibility of making sure that preservice elementary teachers understand the elementary
mathematics content and feel confident enough to teach it
using alternative modes of instruction such as guided
discovery as opposed to direct instruction. In addition
pre-service elementary school teachers should also be
familiarized with the local and national standards as well
as periodicals that will help them as practitioners in the
elementary classroom. Ideally teachers will be able to
find the right balance between what the research suggests
are the best ways of learning and teaching and standards
that suggest what the ideal outcomes ought to be. In the
remainder of this paper, a pre-service elementary teacher
(Michelle Pizzulli: the second author) takes on the
arduous task of balancing the research with the standards
and designing an instructional approach that is also
compatible with her beliefs about the teaching and
learning of mathematics. This narrative is left in the first
person in order to allow the reader to form an
appreciation of the types of decision making involved in
planning a geometry lesson. It also assumes that the
reader is vaguely familiar with the Geometer’s
Sketchpad, a dynamic geometry software that is
commonly found in computer labs at educational
institutions and geo-boards, a useful physical
manipulative to teach geometric concepts.
Designing an Ideal Lesson in Area: Michelle’s journey
What am I trying to teach?
I would like to teach concepts of measurement to fifth
graders. In particular, the students will be able to
calculate the area of a variety of polygons using
geoboards and by utilizing the dynamic geometry
software program Geometer’s Sketchpad (GSP). In
calculating area, students will practice and apply
mathematical skills previously learned such as
multiplication. Although we may not realize it, an
understanding of area and spatial reasoning is very
ZDM 2005 Vol. 37 (5)
applicable in everyday life, such as building a tree fort,
parking a car, or finding room for a new piece of
furniture or rug.
What do the Standards say?
The NCTM emphasizes the importance of geometrical
reasoning at all grade levels. In particular, my lesson
focuses on enabling all students to use visualization,
spatial reasoning and geometric modeling to solve
problems. The students will fulfill the standards by
drawing geometric objects using Sketchpad and use
geometric models to solve problems in other areas of
mathematics, in particular measurement. The activities in
my lesson pose worthwhile mathematical tasks to the
students. These activities engage the students and are
challenging. They develop the student’s mathematical
understanding and skills, and stimulate students to make
connections between area formulas of many different
polygons. Since I ask the students to discover the
formulas on their own, rather than tell them the formulas,
they are involved in mathematical reasoning. While
finding the formulas, students will use a variety of tools
such as Geoboards and Geometer’s Sketchpad to reason
and make connections and solve problems. They will
make conjectures, present solutions, and rely on
mathematical evidence to determine validity of these
solutions. While students work, I will be observing,
listening to and gathering information about the students
to assess what they’re learning. This will ensure that
each student is being challenged, yet not struggling, and
help me decide if any adaptations or changes need to be
made to the lesson.
What does the research say?
In preparing this lesson, I did some research on the
importance of geometry in the curriculum, as well as the
involvement of technology in the teaching of
mathematics, in particular the use of dynamic geometry
software. I found a variety of articles supporting my
beliefs that we must not leave geometry on the back
burner, and that the use of technology can be exciting
while at the same time enhance a student’s understanding
of geometry.
Shannon (2002) discusses the lack of geometry being
taught in schools today. He recognizes it as a big
problem, stating that “currently, there is less geometry
being taught in schools than at any time since the Middle
Ages…”(p.26). Shannon says this could possibly be due
to the perception that geometry “has little relevance in
modern society”(p.26). However, geometry is incredibly
important and applicable in many work situations as well
as in every day life. In his article, Shannon discusses
spatial awareness and understanding, which directly
relates to my topic of study, area. Since we live in a
spatial environment, and much of what we do requires
spatial reasoning, the study of geometry is extremely
important and should not be left behind in today’s
curriculum.
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Scher (1999) poses the question, “Can dynamic geometry
software be used for proofs as well as a demonstration
tool?” At first glance it might appear that programs such
as Geometer’s Sketchpad only provide a demonstration,
leaving the user clueless as to why something occurs in
geometry. Must we just accept a theorem for true without
having any proof to back it up? Scher (1999) suggests
that such dynamic geometry software packages offer the
tools to jumpstart ideas that lead to a proof. They can
demonstrate things (i.e. movement) that a physical model
is not able to show. The use of the tools and functions in
these software programs influence how we reason
mathematically. I used some of these thoughts when
creating my lesson plan on the area of polygons. Rather
than just give the students a formula to accept as true, I
had them come up with the formulas of certain polygons
using Sketchpad. This way they were able to see why
something worked, which in the end results in real
learning, rather than just memorization.
Wilson (1999) discusses the need for teachers to
familiarize themselves with and become comfortable with
new forms of technology for the classroom, in particular
dynamic geometry software. Teachers who have not
been trained on such programs might be afraid or
unwilling to use them. This is unfortunate considering
the benefits these programs have to offer in teaching
important subjects such as geometry. Wilson notes that
many of the activities are quite exciting for teachers, and
most definitely for the students. I can attest to that. I’m
sure my math professor has heard me “wowing” over
something really neat that I’ve just discovered on
Sketchpad. Also, these programs are fairly easy to use,
requiring little help from the teacher. Therefore, he or
she may be available to answer any of the student’s
needs, rather than walking them through it step by step.
In today’s age of increasing technology, it is important to
prepare students to use a variety of computer programs.
However this is only possible if the teachers themselves
are comfortable with the programs. Fortunately, Math
131 is doing a good job in preparing us to bring these
tools into our classrooms. I even felt confident enough to
write up some of my own Sketchpad labs to use in the
classroom.
Clements and Lindquist (2001) state that the geometry
standard is included as one of the five content standards
for mathematics. That means it is just as important as the
others, and it is the teacher’s responsibility to provide
opportunities for learning and using geometry. They talk
about using geometry to investigate, make conjectures
and developing logical arguments to justify conclusions.
This is exactly what my lesson is based on. I ask my
students to explore area, make conjectures on the area of
a variety of polygons, and derive the formulas
themselves. The article discusses the importance of
recognizing how one shape may be made up of other
shapes. My students will see how a parallelogram is
actually made up of a rectangle of the same area. They
will use this information to determine the formula for
area of a parallelogram. They will do several other
activities relating the area of one shape to determine the
area of another. In this way geometry is used as a tool to
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ZDM 2005 Vol. 37 (5)
foster reasoning, for the students will see where the
formula comes from rather than just blindly accepting it.
will try to find the area of some polygons using
geoboards.”
Romagnano (1994) writes about two completely different
methods of teaching. The first is to simply give your
students a problem and tell them exactly how to go about
it, allowing them no room to make inferences of their
own. The second method involves posing a problem to
the class and allowing them to go about it in any way
they wish to arrive at the answer. Romagnano (1994)
discusses how each of these methods has its benefits and
downfalls. The “tell-them” method, while providing the
students the means of arriving at the correct answer,
requires no mathematical reasoning. The students can
perform the task correctly, however they don’t know why
the answer is so. The “ask-them” method did involve
students in the process of the problem, however the
absence of guidelines often left them struggling and
frustrated. Some students weren’t getting much out of
this method, even though they were actively engaged in
the problem. Somewhere, a middle ground must be
reached, where the students are asked to come up with
solutions with some guidelines from the teacher. This is
exactly what I aim to do in my lesson. I do not give the
formulas to the students for memorization, rather I ask
them to tell me the formulas. However, I give them fairly
specific guidelines to follow to help them in the process. I
believe that with the lessons I’ve created, each student
will have the opportunity to think and reason on their
own and be challenged, yet not be left behind.
Hand out geoboards, rubber bands and worksheet with
simple polygons to model. I will have the students find
the area of a square and a rectangle by counting how
many “square units” it is made up of. I will ask them if
they know of a quicker way to do it, then suggest they
multiply one side by an adjacent side and see what they
get. Is it the same? Then I will have them work on more
complex polygons (parallelograms, trapezoids, triangles,
even concave polygons). I will first let them experiment
without telling them how to go about finding the area.
Students may work in small groups at their tables to
brainstorm. I will ask for any ideas, and if nobody gets it
I will model two methods of finding area using the
geoboards. The first, the addition method, involves
breaking up the polygon into smaller pieces in which the
area is easily determined, and then adding up the sum of
those smaller areas. The second, rectangle method,
involves constructing a large rectangle that contains the
entire polygon, finding the area of the rectangle, then
subtracting the number of square units making up the
polygon. I will have the students find the area of a
number of different polygons using both of these
methods, and at this time I will stroll around the
classroom answering questions and providing helpful
suggestions/hints.
The Lesson
A Lesson In Area is a three-day lesson in which the
students will explore the areas of a variety of polygons.
It utilizes the use of technology as well as other
manipulatives in the math classroom. Following the
NCTM Standards, the lesson requires that students use
visualization, spatial reasoning and geometric modeling
to solve a problem. The first day I simply introduce the
concept of area, and we practice finding the area of a
variety of polygons using the geoboards. Students will
discover how to easily find the area of a rectangle and
come up with the formula. This will be the basis for what
we do later in sketchpad. Day two we will go to the lab
and use Geometer’s Sketchpad to discover formulas for
the area of a parallelogram and a triangle. Finally, on day
three, we will continue in the lab, using our previous
knowledge to determine area formulas for a trapezoid and
hexagon. These are the very first Sketchpad labs I’ve
written myself and I am quite proud of them. Here
goes…
Day One
I will introduce the topic of area: “We know that
perimeter is the distance around a closed curve, but what
if we wish to know how much space is inside that region?
The measurement of all the space inside a region is called
area. We use special formulas to find the area of
different types of polygons, but before we get into that we
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Day Two
We will spend this day in the lab. At the beginning of
class I will ask the students if they remember how to
calculate the area of a rectangle. Then I will tell them
that the area of a parallelogram can be found by finding
the area of a rectangle. They will open up Geometer’s
sketchpad and:
Construct a parallelogram and starting with the top left
corner label its vertices clockwise, ABCD.
Now, draw a line perpendicular to line DC that passes
through point A. Also draw a line perpendicular to line
DC that passes through point B. Do you notice how it
creates a rectangle?
Now you can find the measurement by using the formula:
A = base x height. The base is the bottom (or top) side of
the parallelogram and the height is either of the
perpendicular segments you drew in step two (distance
between the two bases). Calculate those measurements,
base and height, and then multiply them. This is the area
of your parallelogram.
Open a new sketch. Construct a triangle. The area of a
triangle can be found using the area of a parallelogram
constructed from two congruent triangles.
Find the midpoint of one side. Select the entire triangle
and mark the midpoint. Go to the Transform menu and
choose Rotate. Rotate the triangle 180 degrees about that
midpoint. What type of polygon does this give you?
Now you can use the formula for the area of a
parallelogram to find the area of the triangle. *Be
careful: The triangle is truly only half the area of the
parallelogram.
Be sure to adjust your formula
accordingly. What is the formula for the area of a
triangle?
ZDM 2005 Vol. 37 (5)
Day Three
We will be in the lab again today, continuing the study of
area using geometer’s sketchpad.
Construct a trapezoid and label it starting with the upper
left hand corner, going clockwise, ABCD.
Draw a segment from point A to C. What does this give
you? Can you find the area of a trapezoid using the area
of two triangles?
Find the area of each triangle. You will need to find the
height by drawing a line through point A that is
perpendicular to DC. Remember, the formula for area of
a triangle is
A = ½ base x height.
Add the areas of the two triangles and the sum is the area
of the entire trapezoid.
What is the formula for area of a trapezoid, then? Use the
distributive property to simplify the formula.
Open a new sketch. Can we find the area of a regular
hexagon by dividing it into congruent triangles? Create
segment AB. Go to the transform menu, mark center A.
Select segment AB and its endpoints. Go back to the
transform menu and choose to rotate the segment 60
degrees. Continue to rotate until you have a total of six
mini-segments, or “spokes”. Connect the endpoints to
make a regular hexagon.
The height of a triangle of a regular polygon is called the
apothem, a. Draw the apothem using a perpendicular
bisector of one of the sides, which passes through the
center of the hexagon.
The area of each triangle is ½ (apothem x side). Use the
measurement of the side which is also an edge of the
hexagon. Find the area of one of the triangles.
Now what do you do to find the area of the entire
hexagon? Remember, there are six triangles contained in
the hexagon. What is the formula for the area of a regular
hexagon?
What would be the formula for a regular n-gon?
Reflections
When reading some of the articles, I realized just how
important geometry is in everyday life. It is vital that we,
as teachers, provide our students with a strong basis in
geometry if they are to be good spatial thinkers. So many
of the things we do rely on basic knowledge of geometry
concepts. In writing my lessons, I wanted my students to
actually learn something, instead of just regurgitating
formulas they have been given. So I asked them to figure
out the formulas on their own, using mathematical
reasoning. I hope that this will make the concept of area
clearer and more permanent. When we just give students
information to memorize and recall, they are not actually
learning anything. I hope that “A Lesson In Area” will
stimulate the students’ brain activity and refine their
skills in problem solving, mathematical thinking and
spatial reasoning. Then, hopefully they will take these
skills with them and apply them to everyday situations.
The lessons presented here I believe are quite challenging
to the students, even those who are gifted. They require a
lot of thinking and problem solving. However, I could
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add some more problems on to the lessons for more
talented math students. I could have them find the area of
a more complex polygon, even a concave polygon. They
could do this by breaking the concave figure up into
recognizable convex polygons and find the area of each
region, then add those up. Or I could alter their lab
activity to allow for more free-thinking. For instance,
rather than give them step by step directions on how to
create a regular hexagon by rotating a segment by 60’
repeatedly, I could ask them how they think we should do
it. I am a huge supporter of having the students learn as
much as possible from themselves or each other, rather
than from the teacher. I feel they get more out of the
lesson if they are actively involved in the process as a
pose to passively taking in information.
Conclusions and Implications
Having presented the active voice of a pre-service teacher
trying to design a lesson that balances the Standards with
mathematics education research several things are
evident. (1) It is evident that balancing the research with
the Standards is no easy task and involves a great deal of
planning, commitment and self reflection on the teachers
part, as evidenced in Michelle’s passionate attempt to
design an ideal lesson. (2) It is also clear that the type of
instructional unit designed is a function of the teacher’s
beliefs about the teaching and learning of mathematics.
Michelle’s lesson reflects her beliefs about mathematics,
as a subject that allows for the discovery of properties
and relationships through personal inquiry. Michelle
would fit Ernest’s (1991) model as person with the
problem-solving view of mathematics.
Michelle’s lesson balances the concerns raised by Hiebert
(1999) regarding the relationship between research and
standards. Hiebert’s (1999) position is that standards are
statements about what is most valued (p.4) and he further
states that research can inform the debate. To Michelle
the Standards conveyed the message that mathematics
should be taught with an emphasis on understanding, an
interpretation that was compatible with her beliefs about
the teaching and learning of mathematics. The survey of
research indicated to her that geometry had numerous
real-world applications, which was again harmonious
with her problem-solving approach to teaching
mathematics.
It must be pointed that Standards are open to individual
interpretation and there is always the possibility that
some teachers might interpret them as teaching
procedures. Since research has informed us about what
students do not comprehend in geometry and eventually
in proof writing, it would make more sense to guide
students into discovering geometry and proof the way
mathematicians do. Deductive proof is the outcome of a
process and not the starting point (Fawcett, 1938, Senk,
1985, Usiskin, 1987).
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The idea of building up directly on students’ knowledge
as they progress from the elementary to higher levels, as
espoused by the Standards is brilliant. The authors hope
that elementary classroom teachers will find the right
balance between research and standards as they help
their students further their ability to read, write and
discuss mathematics and provide them with an
opportunity for discovery and invention.
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Authors
Prof. Dr. Bharath Sriraman
436
ZDM 2005 Vol. 37 (5)
Editor, The Montana Mathematics Enthusiast
http://www.montanamath.org/TMME
Dept. of Mathematical Sciences,
The University of Montana,
Missoula, MT 59812,
USA.
E-mail: sriramanb@mso.umt.edu
Michelle Pizzulli
The University of Montana